Content

• To step
• Method 1 of 2: Root pulling with prime factors
• Method 2 of 2: Finding square roots without a calculator
• With long division
• Understand the procedure
• Tips
• Warnings

Before the advent of calculators, both students and professors had to calculate square roots with pen and paper. At the time, various techniques were developed to tackle this sometimes difficult job, some of which give a rough estimate and others calculate the exact value. Read on to learn how to find the square root of a number in a few easy steps.

To step

Method 1 of 2: Root pulling with prime factors

1. Divide your number into power factors. This method uses the factors of a number to find the square root of a number (depending on the number, it can be an exact answer or an estimate). The factors of a given number are any sequence of numbers that are multiplied together to form that particular number. For example, you could say that the factors of 8 are equal to 2 and 4 because 2 × 4 = 8. Perfect squares, on the other hand, are integers that are the product of other integers. For example, 25, 36, and 49 are perfect squares because they are equal to 5, 6, and 7, respectively. Second power factors, as you will have understood, are factors that are also perfect squares. To find a square root using prime factors, first try to divide the number into its second-power factors.
   • Take the following example. We're going to find the square root of 400. To begin with, we divide the number into power factors. Since 400 is a multiple of 100, we know it is evenly divisible by 25 - a perfect square. Quick rote tells us that 400/25 = 16.16 also happens to be a perfect square. So the cube factors of 400 are 25 and 16 because 25 × 16 = 400.
   • We write this as: Sqrt (400) = Sqrt (25 × 16)
2. Take the square roots of your second power factors. The product rule of square roots states that for any given number a and b, Sqrt (a × b) = Sqrt (a) × Sqrt (b). Because of this property, we can now take the square roots of the squares factors and multiply them together to get the answer.
   • In our example, we take the square roots of 25 and 16. See below:
     • Sqrt (25 × 16)
     • Sqrt (25) × Sqrt (16)
     • 5 × 4 = 20
3. If your number can't be factored up perfectly, simplify it. In reality, the numbers you want to determine the square roots of won't be nice rounded numbers with nice squares like 400. In these cases, it may not be possible to get an integer answer. Instead, using all the power factors you can find, you can determine the answer as a smaller, easier-to-use square root. You do this by reducing the number to a combination of power factors and other factors, and then simplifying it.
   • We take the square root of 147 as an example. 147 is not the product of two perfect squares, so we cannot get a nice integer value. But it is the product of a perfect square and another number - 49 and 3. We can use this information to write our answer in the simplest terms:
     • Sqrt (147)
     • = Sqrt (49 × 3)
     • = Sqrt (49) × Sqrt (3)
     • = 7 × Sqrt (3)
4. Simplify, if necessary. Using the square root in the simplest terms, it is usually fairly easy to get a rough estimate of the answer by estimating the remaining square roots and multiplying them. One way to improve your guesses is to find the perfect squares on either side of the number in your square root. You know that the decimal value of the number in your square root is somewhere in between these two numbers, so your guess will have to be between these numbers as well.
   • Let's return to our example. Since 2 = 4 and 1 = 1, we know that Sqrt (3) is between 1 and 2 - probably closer to 2 than 1. We estimate that 1.7. 7 × 1.7 = 11,9. If we check this with the calculator, we see that we are pretty close to the answer: 12,13.
     • This also works for the larger numbers. For example, sqrt (35) is roughly between 5 and 6 (probably closer to 6). 5 = 25 and 6 = 36.35 is between 25 and 36, so the square root will be between 5 and 6. Since 35 is just below 36, we can say with some confidence that the square root of it just is less than 6. Checking with a calculator gives us an answer of about 5.92 - we were right.
5. Alternatively, as a first step, you can simplify the number to the least common multiple. Searching for power factors is not necessary if you can easily find prime factors of a number (factors that are also prime numbers at the same time). Write the number in terms of least common multiples. Then search between your factors for matching pairs of prime numbers. When you find two prime factors that match, remove them from the square root and place a of these numbers outside the square root sign.
   • For example, we determine the square root of 45 using this method. We know that 45 = 9 × 5 and that 9 = 3 × 3. So we can write the square root like this: Sqrt (3 × 3 × 5). Simply delete the 3's and place a 3 outside the square root to get a simplified square root: (3) Sqrt (5). Now you can easily make an estimate.
   • A final example; we determine the square root of 88:
     • Sqrt (88)
     • = Sqrt (2 × 44)
     • = Sqrt (2 × 4 × 11)
     • = Sqrt (2 × 2 × 2 × 11). We have several 2's in our square root. Since 2 is prime, we can remove a pair and place a 2 outside the root.
     • = Our square root in simplest terms is (2) Sqrt (2 × 11) or (2) Sqrt (2) Sqrt (11). Now we can approach Sqrt (2) and Sqrt (11) and find an approximate answer, if we wanted to.

Method 2 of 2: Finding square roots without a calculator

With long division

1. Divide the digits of your number into pairs. This method is similar to long division, which allows you to use the exact square root of a number digit by digit. Although not essential, breaking a number into workable pieces can make solving easier, especially if it is long. First draw a vertical line dividing the work area into 2 areas, then a shorter line near the top of the right area, dividing it into a smaller top part and a larger part below. Then divide the number into pairs of numbers, starting from the decimal point. Under this rule, 79520789182.47897 becomes "7 95 20 78 91 82.47 89 70". Write this number in the top left area.
   • As an example, let's calculate the square root of 780.14. Divide your work space as above and write "7 80, 14" in the top left corner. It's okay if there is only one number on the far left, instead of two. You then write the answer (the square root of 780.14) at the top of the right area.
2. Find the largest integer n whose square is less than or equal to the left-most digit or number. Find the greatest square that is less than or equal to this number, and then find the square root of this square. This number is n. Write that in the top right area and write the square of n in the bottom quadrant of that area.
   • In our example, the left-most digit is the number 7. Since we know that 2 = 4 ≤ 7 3 = 9, we can say that n = 2 because this is the largest integer whose square is less than or equal to 7. Write 2 in the top right quadrant. This is the first digit of the answer. Write 4 (the square of 2) in the lower right quadrant. This number is important for the next step.
3. Subtract the number you calculated of the leftmost digit or number. As with long division, the next step is to subtract the square from the number we just used for the calculation. Write this number under the leftmost number and subtract them. Write the answer below.
   • In our example, we write a 4 under 7 and subtract it. This gives 3 in response.
4. Move the next number down. Place this next to the value you found in the previous edit. Multiply the number at the top right by two and write it down at the bottom right. Leave space next to the number you just wrote down for the multiplication sum that you will do in the next step. Write here "_ × _ =" ".
   • In our example, the next number is "80". Write "80" next to the 3 in the left quadrant. Then multiply the number at the top right by 2. This number is 2, so 2 × 2 = 4. Write down "" 4 "" at the bottom right, followed by _×_=.
5. Enter the numbers on the right. In the blank space of the sum (right), enter the largest integer that makes the result of the multiplication sum on the right less than or equal to the current number on the left.
   • In our example, we enter 8, and this gives 4 (8) × 8 = 48 × 8 = 384. This is greater than 380. So 8 is too big, but 7 probably isn't. Fill in 7 and solve: 4 (7) × 7 = 329. 7 is good because 329 is less than 380. Write 7 at the top right. This is the second digit in the square root of 780.14.
6. Subtract the number you just calculated from the current number on the left. So you subtract the result of the multiplication on the right from the current answer on the left. Write your answer directly below it.
   • In our example, we subtract 329 from 380, and this gives 51 as result.
7. Repeat step 4. Move the next pair of numbers down from 780.14. When you arrive at a comma, write that comma in the answer to the right. Then multiply the top right number by 2 and write the answer next to ("_ × _") as above.
   • In our answer we now write a comma because we also encounter this in 780.14. Move the next pair (14) down the left quadrant. 27 x 2 = 54, so we write "54 _ × _ =" in the lower right quadrant.
8. Repeat steps 5 and 6. Find the largest number that gives an answer that is less than or equal to the current number on the left. Solve.
   • In our example, 549 × 9 = 4941, which is less than or equal to the number on the left (5114). 549 × 10 = 5490, which is too high, so 9 is our answer. Write 9 as the next top right number and subtract the result of the multiplication from the left number: 5114 -4941 = 173.
9. To make the result accurate, repeat the previous procedure until you find the answer with the number of decimal places (hundredths, thousandths) you need.

Understand the procedure

1. Consider the number whose square root you want to calculate as the area S of a square. Since the area of ​​a square is L, where L is the length of one of its sides, so by finding the square root of your number, you try to calculate the length L of the side of that square.
2. Give each digit of your answer a letter. Enter the variable A as the first digit of L (the square root we are trying to calculate). B is the second digit, C the third, and so on.
3. Give a letter to each "pair of numbers" of the number you start with. Give the variable S_a to the first pair of digits in S (the initial value), S._b to the second pair of digits, etc.
4. Understand the relationship between this method and long division. This method of finding a square root is essentially a long division, where you divide the initial value by its square root and "give" the square root as the answer. As with long division, where you are only interested in the next digit at a time, you are only interested in the next two digits at a time (which correspond to the next digit of the square root).
5. Find the largest number whose square is less than or equal to S._a is. The first digit A in our answer is then the largest integer whose square is not greater than S._a (A such that A² ≤ Sa (A + 1) ²). In our example, S_a = 7, and 2² ≤ 7 3², so A = 2.
   • Note that if you divide 88962 by 7 using long division, the first step is equal: you first deal with the first digit of 88962 (8) and you want the largest digit multiplied by 7 that is less than or equal to 8. Essentially you determine d such that 7 × d ≤ 8 7 × (d + 1). In this case, d equals 1.
6. Visualize the square you want to find the area of. Your answer, the square root of the initial value, is L, which describes the length of a square with area S (the initial value). The values ​​for A, B, and C represent the digits in the value L. Another way of saying this is that for a 2-digit answer, 10A + B = L, and for a 3-digit answer, 100A + 10B + C = L, and so on.
   • In our example (10A + B) ² = L = S = 100A² + 2 × 10A × B + B². Remember that 10A + B represents our answer L along with B in the units position, and A in the tens position. For example, if A = 1 and B = 2, then 10A + B is the number 12. (10A + B) ² is the area of ​​the entire square, while 100A² is the area of ​​the largest inner square, B² is the area of ​​the smallest square and 10A × B is the area of ​​each of the remaining rectangles. Through this long, complicated procedure, we can find the area of ​​the entire square by adding the areas of the squares and rectangles that are part of it.
7. Subtract A² from S._a. Bring a pair of numbers (S._b) down from the number S. S._a S._b is almost the total area of ​​the square, from which you just subtracted the area of ​​the largest inner square. The remainder is, say, the number N1, which we obtained in step 4 (N1 = 380 in our example). N1 equals 2 × 10A × B + B² (the area of ​​the 2 rectangles plus the area of ​​the small square).
8. Look at N1 = 2 × 10A × B + B², also written as N1 = (2 × 10A + B) × B. In our example, you already know N1 (380) and A (2), so you now need to find B. B probably isn't an integer, so you have to actually Find the largest integer B, such that (2 × 10A + B) × B ≤ N1. So now you have: N1 (2 × 10A + (B + 1)) × (B + 1).)
9. Solve the equation. To solve this equation, multiply A by 2, shift it to the ten (multiply by 10), put B in the units, and multiply the result by B. In other words, (2 × 10A + B) × B. This is exactly what you do when you write "N_ × _ =" (with N = 2 × A) in the lower right quadrant in step 4. In step 5 you determine the largest integer B that fits below the line, so (2 × 10A + B) × B ≤ N1.
10. Subtract the area (2 × 10A + B) × B from the total area. This gives the area S- (10A + B) ² that you have not yet taken into account (and that you use to calculate the following numbers in the same way).
11. To calculate the next digit C, repeat the procedure. Move the next pair of numbers from S down (S_c) to get N2 to the left, and look for the largest C so that you now have: (2 × 10 × (10A + B) + C) × C ≤ N2 (equal to twice the two-digit number "AB" followed by "_ × _ =" Now determine the largest number you can enter here, which will give you an answer that is less than or equal to N2.

Tips

• Moving the comma by two places (a factor of 100) moves the comma in the corresponding square root by one place (a factor of 10).
• In the example, 1.73 could be considered "remainder": 780.14 = 27.9² + 1.73.
• This method works for any number system, not just the decimal (decimal) system.
• Feel free to place the calculations where you want. Some people write it above the number they want to calculate the square root of.
• An alternative method is the following: √z = √ (x ^ 2 + y) = x + y / (2x + y / (2x + y / (2x + ...))). For example, to calculate the square root of 780.14, take the integer whose square is closest to 780.14 (28), so = 780.14, x = 28, and y = -3.86. Filling in and estimating gives us x + y / (2x) and this gives (simplified terms) 78207/2800 or about 27.931 (1); the following term, 4374188/156607 or about 27.930986 (5). Each term adds about 3 decimal places of precision to the previous one.

Warnings

• Make sure to divide the number into pairs from the decimal point. Dividing 79520789182.47897 as "79 52 07 89 18 2,4 78 97 "gives an incorrect result.